In Chapter 2 we briefly introduced the concept of a marketimposed "tradeoff" between risk and return for securities  that is, the higher the risk of a security, the higher the expected return that must be offered the investor. We made use of this concept in Chapter 3. There we viewed the value of a security as the present value of the cashflow stream provided to the investor, discounted at a required rate of return appropriate for the risk involved. We have, however, purposely postponed until now a more detailed treatment of risk and return. We wanted you first to have an understanding of certain valuation fundamentals before tackling this more difficult topic.

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The return from holding an investment over some period  say, a year  is simply any cash payments received due to ownership, plus the change in market price, divided by the beginning price. You might, for example, buy for $100 a security that would pay $7 in cash to you and be worth $106 one year later. The return would be ($7 + $6)/$100 = 13%. Thus return comes to you from two sources: income plus any price appreciation (or loss in price).

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Most people would be willing to accept our definition of return without much difficulty. Not everyone, however, would agree on how to define risk, let alone how to measure it.

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Using Probability Distributions to Measure Risk 
As we have just noted, for all except riskfree securities the return we expect may be different from the return we receive. For risky securities, the actual rate of return can be viewed as a random variable subject to a probability distribution. Suppose, for example, that an investor believed that the possible oneyear returns from investing in a particular common stock were as shown in the shaded section of Table, which represents the probability distribution of oneyear returns.

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Expected Return and Standard Deviation 
The expected return, R, is
(5.2)
where Rj is the return for the ith possibility, P; is the probability of that return occurring, and n is the total number of possibilities. Thus the expected return is simply a weighted average of the possible returns, with the weights being the probabilities of occurrence. For the distribution of possible returns shown in Table 5.1, the expected return is shown to be 9 percent.

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The standard deviation can sometimes be misleading in comparing the risk, or uncertainty, surrounding alternatives if they differ in size. Consider two investment opportunities, A and B, whose normal probability distributions of oneyear returns have the following characteristics:

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Just when you thought that you were safely immersed in the middle of a finance chapter, you find yourself caught up in a time warp, and you are a contestant on the television game show Let's Make a Deal. The host, Monty Hall, explains that you get to keep whatever you find behind either door #1 or door #2. He tells you that behind one door is $10,000 in cash, but behind the other door is a "zonk," a used tire with a current market value of zero. You choose to open door #1 and claim your prize. But before you can make a move, Monty says that he will offer you a sum of money to call off the whole deal.

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So far, we have focused on the risk and return of single investments held in isolation. Investors rarely place their entire wealth into a single asset or investment. Rather, they construct a portfolio or group of investments. Therefore we need to extend our analysis of risk and return to include portfolios.

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Portfolio Risk and the Importance of Covariance 
Although the portfolio expected return is a straightforward, weighted average of returns on the individual securities, the portfolio standard deviation is not the simple, weighted average of individual security standard deviations. To take a weighted average of individual security standard deviations would be to ignore the relationship, or covariance, between the returns on securities. This covariance, however, does not affect the portfolio's expected return.

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The concept of diversification makes such common sense that our language even contains everyday expressions that exhort us to diversify ("Don't put all your eggs in one basket"). The idea is to spread your risk across a number of assets or investments. While pointing us in the right direction, this is a rather naive approach to diversification. It would seem to imply that investing $10,000 evenly across 10 different securities makes you more diversified than the same amount of money invested evenly across 5 securities. The catch is that naive diversification ignores the covariance (or correlation) between security returns. The portfolio containing 10 securities could represent stocks from only one industry and have returns that are highly correlated. The 5stock portfolio might represent various industries whose security returns might show low correlation and, hence, low portfolio return variability.

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